An Example of Constructive Defining: from a Golden Rectangle To

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An example of constructive defining: From a golden rectangle to golden quadrilaterals & beyond: Part 2 MICHAEL DE VILLIERS University of Stellenbosch, South Africa e-mail: [email protected] Dynamic Geometry Sketches: http://dynamicmathematicslearning.com/JavaGSPLinks.htm Keywords: golden ratio, golden isosceles trapezium, golden kite, golden hexagon

This article continues the investigation started by the author in the March 2017 issue of At Right Angles, available at: http://teachersofindia.org/en/ebook/golden-rectangle-golden-quadrilaterals-and-beyond-1 It is part of a paper which continues the investigation started in De Villiers (in press) about constructively defining various golden quadrilaterals analogously to the famous golden rectangle so that they exhibit some aspects of the golden ratio phi. Constructive defining refers to the defining of new objects by modifying or extending known definitions or properties of existing objects. In the first part of the paper, different possible definitions were proposed for the golden rectangle, golden rhombus and golden parallelogram, and they were compared in terms of their properties as well as ‘visual appeal’.

In this part of the paper, we shall first look at possible definitions for a golden isosceles trapezium as well as a golden kite, and later, at a possible definition for a golden hexagon. Constructively Defining a ‘Golden Isosceles Trapezium’ From the first construction, it follows that triangle DXC is equilateral, and therefore XC = How can we constructively define a ‘golden’ isosceles trapezium? Again, there are a. Hence, BC/AD = (phi + 1)/phi, which is well known to also equal phi1. This result several possible options. It seems natural though to first consider constructing a golden isosceles trapeziumtogether ABCD with in twothe differentsimilarity ways of fromisosceles a golden triangles parallelogram AED and (ABXD CEB in, further implies that the 1st case, and CEAXCD/EA in= theBE /2EDnd case = phi.) with In another acute words, angle notof 60° only as areshown the inparallel Figure sides8. In in the golden ratio, the first constructionbut the shown, diagonals this amountalso divides to defining each other a golden in the isosceles golden ratio. trapezium Quite as nice! an isosceles trapezium ABCDIn the with second AD // case,BC, anglehowever, ABC AD= 60°,/BC and= phi/(phi the (shorter – 1) parallel)= (phi + 1) = phi squared. side AD and ‘leg’Also AB notein the in golden the second ratio phi case,. in contrast to the first, it is the longer parallel side AD that is in the golden ratio to the ‘leg’ AB, and the ‘leg’ AB is in the golden ratio with the An example of constructive defining: shorter side BC. So the pairs of sides of this golden isosceles trapezium form a geometric progression from the shorter to the longest side, which is quite nice too! From a GOLDEN Subdividing the golden isosceles trapezium in the first case in Figure 8, like the golden parallelogram in Figure 5, by respectively constructing a rhombus or two TechSpace Figure 8: Constructing a golden isosceles trapezium in two ways RECTANGLE to GOLDEN equilateralFigure 8. tr Constructingiangles at a golden the ends,isosceles clearlytrapezium indoes two waysnot produce an isosceles trapezium similar to the original. In this case the parallel sides (longest/shortest) of the obtained isosceles the 2nd case) with an acute angle of 60° as shown trapezium are also in the ratio (phi + 1), and is in Figure 8. In thetrapezium first construction are also shown, in thisthe ratiotherefore (phi + in1), the and shape is oftherefore the second in type the in shape Figure of the second type in QUADRILATERALS amounts to defining a golden isosceles trapezium 8. The rhombus formed by the midpoints of the as an isosceles trapeziumFigure ABCD 8. The with rhombus AD // BC, formedsides by of thethe firstmidpoints golden isosceles of the trapezium sides of is thealso first golden isosceles angle ABC = 60°, andtrapezium the (shorter is alsoparallel) not side any of thenot previouslyany of the previously defined defined ‘golden’ ‘golden’ rhombi. rhombi. AD and ‘leg’ AB in the golden ratio phi. and Beyond Part 2 With reference to the first construction, we could From the first construction,With it follows reference that triangle to thedefine first the construction, golden isosceles wetrapezium could without define any the golden isosceles DXC is equilateral,trapezium and therefore without XC = a. Hence,any reference reference to tothe the 60° 60° angle as asan isoscelesan isosceles trapezium trapezium ABCD with MICHAEL DE VILLIERS This article continues the investigation started by the author in the March BC/AD = (phi + 1)/phi, which is well known to ABCD with AD // BC, and AD/AB = phi = BC/AD 1 2017 issue of At Right Angles, available at: http://teachersofindia.org/en/ also equal phi . ThisAD result// BC together, and ADwith/ ABthe = phi =as BC shown/AD in as Figure shown 9. in Figure 9. similarity of isosceles triangles AED and CEB, ebook/golden-rectangle-golden-quadrilaterals-and-beyond-1 The focus further implies that CE/EA = BE/ED = phi. In of the paper is on constructively defining various golden quadrilaterals other words, not only are the parallel sides in the analogously to the famous golden rectangle so that they exhibit some golden ratio, but the diagonals also divide each aspects of the golden ratio phi. Constructive defining refers to the defining other in the golden ratio. Quite nice! of new objects by modifying or extending known definitions or properties In the second case, however, AD/BC = phi/ of existing objects. In the first part of the paper in De Villiers (2017), (phi – 1) = (phi + 1) = phi squared. Also note in different possible definitions were proposed for the golden rectangle, golden the second case, in contrast to the first, it is the rhombus and golden parallelogram, and they were compared in terms of longer parallel side AD that is in the goldenFigure ratio 9: AlternativeFigure definition 9. Alternative for definition first forgolden first golden isosceles trapezium their properties as well as ‘visual appeal’. to the ‘leg’ AB, and the ‘leg’ AB is in the golden isosceles trapezium ratio with the shorter side BC. So the sides of In this part of the paper, we shall first look at possible definitions for a this golden isoscelesHowever, trapezium formthis a isgeometric clearly notHowever, as convenient this is clearly a defininot as convenienttion as sucha a choice of definition golden isosceles trapezium as well as a golden kite, and later, at a possible progression from the shortest to the longest side, definition, as such a choice of definition requires requires again use of the cosineagain formula the use of to the show cosine that formula it implies to show that angle ABC = 60° (left definition for a golden hexagon. which is quite nice too! that it implies that angle ABC = 60° (left to Subdividing the goldento the isosceles reader trapezium to verify). in the As theseen reader earlier, to verify). stating As seen one earlier, of stating the angles and an appropriate Constructively Defining a ‘Golden Isosceles Trapezium’ first case in Figure 8, like the golden parallelogram one of the angles and an appropriate golden How can we constructively define a ‘golden’ isosceles in Figure 5, by respectively constructing a ratio of sides or diagonals in the definition, 1 trapezium? Again, there are several possible options. It rhombus or two equilateral Keep in triangles mind that at thephi ends,is defined assubstantially the solution simplifies to the quadratic the deductive equation structure. phi2 – phi – 1 = 0. From this, it seems natural though, to first consider constructing a golden clearly does not producefollows an that isosceles phi = trapezium(phi + 1)/phi, phiThis = 1/(phi illustrates -1), phi/(phi the important +1) = phieducational + 1, or phipoint2 = phi + 1. isosceles trapezium ABCD in two different ways from a similar to the original. In this case the parallel that, generally, we choose our mathematical golden parallelogram (ABXD in the 1st case, and AXCD in sides (longest/shortest) of the obtained isosceles definitions for convenience and one of the criteria 1

Keywords: Golden ratio, golden isosceles trapezium, golden kite, golden 1 Keep in mind that phi is defined as the solution to the quadratic equation phi2 – phi – 1 = 0. From hexagon, golden triangle, golden rectangle, golden parallelogram, this, it follows that phi = (phi + 1)/phi, phi = 1/(phi -1), phi/(phi +1) = phi + 1, or phi2 = phi + 1. golden rhombus, constructive defining

74 At Right Angles | Vol. 6, No. 2, August 2017 At Right Angles | Vol. 6, No. 2, August 2017 75

At Right Angles | Vol. 6, No. 2, August 2017 PB An example of constructive defining: From a golden rectangle to golden quadrilaterals & beyond: Part 2 MICHAEL DE VILLIERS University of Stellenbosch, South Africa e-mail: [email protected] Dynamic Geometry Sketches: http://dynamicmathematicslearning.com/JavaGSPLinks.htm Keywords: golden ratio, golden isosceles trapezium, golden kite, golden hexagon

74 At Right Angles | Vol. 6, No. 2, August 2017 At Right Angles | Vol. 6, No. 2, August 2017 75

At Right Angles | Vol. 6, No. 2, August 2017 PB golden ratio of sides or diagonals in the definition, substantially simplifies the deductive structure.golden This ratio illustrates of sides orthe diagonals important in educationalthe definition, point substantially that, generally, simplifies we thechoose deductive our mathematicalstructure. definitionsThis illustrates for convenienceforthe ‘convenience’ important andis educational the oneease byof which the point criteriathe otherthat, for generally, ‘convenience’A fourth we way choose to defineis the our and conceptualize a golden ease mathematicalby which the othedefinitionsr properties propertiesfor convenience can can be be derived derived and from fromone it. of it. the A fourthcriteria way for isoscelesto‘convenience’ define trapezium and conceptualize couldis the be to start aagain golden with isosceles trapezium could be to start a golden triangle ABX, but this time to translate ease by which the other properties can be derived from againit. with a goldenit with thetriangle vector BXABX along, but its this‘base’ time to produce to translate it with the vector BX along its a golden isosceles trapezium ABCD as shown in ‘base’ to produceFigure a golden11. In this isosceles case, since trapezium the figure is ABCD made as shown in Figure 11. In this case, since the figureup is of made3 congruent up of golden 3 congruent triangles, itgolden follows thattriangles, it follows that AB/AD = phi, AB/AD = phi, and BC = 2AD (and therefore its FigureFigure 13: Fir13. stFirst case case of of golden golden kite kite and BC = 2ADdiagonals (and therefore also divide its each diagonals other in the als ratioo divide each other in the ratio 2 to 1). 2 to 1). Though one could maybe argue that the first case of a golden isosceles trapezium A fourth way to define and conceptualize Thougha golden one isosceles could maybe trapezium argue that thecould first be case to start in Figure 8 is oftoo a golden‘broad’ isosceles and trapeziumthe one in FigureFigure 8 is11 too is too ‘tall’ to be visually appealing, again with a golden triangle ABX, but this time to translate it with the vector BX along its ‘broad’ and the one in Figure 11 is too ‘tall’ to be Figure 10. Third golden isosceles trapeziumthere is little visually appealing, different there betwe is littleen visually the differentone in Figure 10 and the second case in Figure‘base’ 10: toThird produce golden a golden isosceles isosceles trapezium trapezium ABCD as shown in Figure 11. In this case, Figure 8. However,between all the four one incases Figure or 10 types and the have second interesting case mathematical properties, and AnotherFiguresince completelythe 10: figure Third different isgolden made way isoscelesup to ofdefine 3 congruent trapeziumand in golden Figure 8. triangles, However, allit fourfollows cases orthat types AB have/AD = phi, Another completely different way to define and conceptualizedeserve to be a known. interestinggolden mathematicalisosceles properties, and deserve conceptualizeand BC = a2 goldenAD (and isosceles therefore trapezium its isdiagonals to also divide each other in the ratio 2 to 1). Another completely differentagain use way a golden to triangle.define As andshown conceptualize in Figure toa be golden known. isosceles Figure 14. Second case of golden kite Figure 14: Second case of golden kite trapezium is to again use a golden10, by reflectingtriangle.Though a Asgolden one shown trianglecould in ABCmaybe Figure in the argue 10, by that reflecting the first acase golden of a golden isosceles trapezium trapezium is to again use aperpendicular golden triangle. bisector ofAs one shown of its ‘legs’ in Figure BC, 10, by reflecting a golden triangle ABC in the perpendicular bisector of one of its ‘legs’ BC, produces a ‘golden ConstructivelyAnother way Definingmight be aagain ‘Golden to defineKite’ the pairs one,of angles and is intherefore the golden perhaps kite a littleto be more in the visually producesin Figure a ‘golden 8 is isosceles too ‘broad’ trapezium’ and where the onethe in Figure 11 is too ‘tall’ to be visually appealing, triangle ABC in the perpendicular bisector of one of its ‘legs’ BC, produces a ‘golden pleasing. This observation, of course, also relates isosceles trapezium’ where theratio ratioBC/AB BCis phi,/AB and is the phi, acute and ‘base’ the angle acute is ‘base’ angle is 72°. Againgolden there ratio are severalas shown possible in waysFigure in which14. Determining x from this geometric progression, there is little visually different between the one in Figure 10 and the second case in to constructively define the concept of a ‘golden to the ratio of the diagonals, which in the first isosceles trapezium’ where72°. theMoreover, ratio sinceBC /angleAB isBAD phi, = 108° and and the angle acute ‘base’ angle is 72°. 360 kite’.rounded An easy off way to oftwo constructing decimals, (andgives: defining) x  case 45 is .2.4084. (roundedOf special off interest to 2 decimals) is that thewhile in Moreover, since angle BAD =ADB 108° = 36°, and it follows angle that ADC angle =ABD 36°, is also it follows36°. that angle ABD is 2 Figure 8. However, all four cases or types have interesting mathematical properties, and 1 2the case in Figure 14, it is 1.84 (rounded off to Moreover, since angle BADHence, = 108°AD = ABand (= DCangle), and ADC therefore = 36°, the two it follows that angle ABD is one might be to again start with a golden triangle 2 decimals), and hence the latter is closer to the also 36°. Hence, AD = AB (=parallel deserveDC ),sides and to are be thereforealso known. in the golden the tworatio, andparallel as sides are also in the andangles construct at B anand equilateral D work triangleout to beon preciselyits base as equal to 120°. This golden kite looks a little golden ratio phi. also 36°. Hence, AD = ABwith (= the DC preceding), and case, therefore the diagonals the thereforetwo parallel sides are also in the shown in Figure 13. Since AB/BD = phi, it follows golden ratio, and as with the preceding case, the diagonals therefore also divide each ‘fatter’ than the precedingBD BC convex one, and is therefore perhaps a little more visually also divide each other into the golden ratio. Of immediately that since =  by construction, To define a golden kite that is hopefully even golden ratio, and as with the preceding case, the diagonals therefore also divideFigure each 12: Golden isosceles trapezia of type 3 AB to BC is also in the golden ratio. Notice that other into the golden ratio. Ofinterest interest also, alsois to note is tothat note the diagonals that the AC diagonals and AC and DB each pleasing. This observation, of course, also relatesmore to the visually ratio appealing of the diagonals, than the previous which intwo, I DB each respectively bisect the ‘base’ angles2 at C the same construction applies to the concave case, other into the golden ratio. Of interest also is to note that the diagonals AC and DB each the first case is 2.40 (rounded off to 2 decimals) nextwhile thought in the ofcase defining in Figure a ‘golden 14, it kite’is 1.84 as shown and B. 2 but is probably not as ‘visually pleasing’ as the 3 respectively bisect the ‘base’ angles at 2B and C. One more argument towards perhaps slightly favoring the golden isosceles trapezium, in Figure 15, namely, as a (convex ) kite with respectively bisect the ‘base’ angles at B and C. convex(rounded case. off to 2 decimals), and hence the latter isboth closer its sides to the and golden diagonals ratio in phi. the golden ratio. defined and constructed in Figure 9, might be that it appears in both the regular convex Another way might be again to define the pairs To define a golden kite that is hopefully even more visually appealing than the pentagon as well as the regular star pentagon as illustrated in Figure 12. of angles in the golden kite to be in the golden previous two, I next thought of defining a ‘golden kite’ as shown in Figure 15, namely, as ratio as shown in Figure 14. Determining x from 3 Constructively Defining a ‘Golden Kite’ thisa (convexgeometric) progression,kite with both rounded its sides off andto two diagonals in the golden ratio. Figure 12. Golden isosceles trapezia of type 3 decimals, gives: FigureAgain 12: there Golden are isoscelesseveral possible trapezia ways of type in which3 to constructively define the concept of a One more argument towards perhaps slightly 360° x = = 45.84° 2 One more argument towards‘golden perhaps kite’. slightly Anfavoring easy favorin the way golden gof the constructingisosceles golden trapezium, isosceles (and defined defining) trapezium, one might be to again start with 1+φ + 2φ and constructed in Figure 10, might be that it defined and constructed in Figurea golden 9, trianglemight be and that construct it appears an inequilateral both the regulartriangle convexon its base as shown in Figure 13. appears in both the regular convex pentagon as Of special interest is that the angles atFi Bgure and 15:D Third case: Golden kite with sides and diagonals in the golden ratio FigureFigure 11: 11:A fourth A fourth golden golden isosceles isosceles trapezium trapezium well as the regular star pentagon as illustrated in Figure 15. Third case: Golden kite with sides and pentagonFigure 11. asA fourth well golden as the isosceles regular trapeziumSince star AB pentagon/BD = phi, as illustratedit follows immediatelyin Figure 12. that since BD = BC by construction, ABwork to out BC to be precisely equal to 120°. This golden diagonals in the golden ratio 1 Figure 12. �� 1 kite3 looks a little ‘fatter’ than theThough preceding one convex can drag a dynamically constructed kite in dynamic geometry with sides is also in the golden ratio. Notice that the same construction applies to the concave For case, the sake of brevity we shall disregard the concave case here. Constructively Defining a ‘Golden Kite’ constructed in the golden ratio so that its diagonals are approximately also in the golden 2 In De Villiers (2009, p. 154-155; 207) a general isoscelesbut trapezium is probably with three adjnotacent as sides ‘visually equal is called apleasing’ trilateral trapezium, as andthe the convex property case. thatAgain a pair of adjacent,there congruentare several angles are bisectedpossible by the diagonalsways is inalso mentioned.which Alsoto see:constructively http://dynamicmathematicslearning.com/quad- define the concept of a ratio, making an accurate construction required the calculation of one of the angles. At tree-new-web.html 3 For the sake of brevity we shall disregard the concave case here. 2 2 ‘golden kite’. An easy way of constructing (and defining) one might be to again start with first I again tried to use the cosine rule, since it had proved effective in the case of one In De In Villiers De Villiers (2009, (2009, p. 154 p.- 155;154- 155;207) 207) a general a general isosceles isosceles trapezium trapezium with with three three adjacent adjacent sides sides equal equal is is calledcalled a trila ateral trila teraltrapezium trapezium, and76, theand property atheAt golden Right property Angles that triangle thata| Vol.pair a 6, pair No.ofand adjacent,2, of August constructadjacent, 2017 congruent congruent an equilateral angles angles are are trianglebisected bisected byon by the its the base as shown in Figure 13. golden parallelogram as well asAt Rightone Anglesisosceles | Vol. 6,trapezium No. 2, August case, 2017 but77 with no success. diagonalsdiagonals is also is mentioned.also mentioned. Also Also see: see:http://dynamicmathematicslearning.com/quad http://dynamicmathematicslearning.com/quad-tree-tree-new-new-web.html-web.html Eventually switching strategies, and assuming AB = 1, applying the theorem of Since AB/BD = phi, it follows immediately that since BD = BC by construction, AB to BC Pythagoras to the right triangles ABE and ADE gave the following:

is also in the golden ratio. Notice that the same construction applies to the concave case, 2 At Right Angles | Vol. 6, No. 2, August 2017 PB 2 BD y  1 but is probably not as ‘visually pleasing’ as the convex case. 4 2

2 BD 2 2  (BD  y)   . 4 2 Solving for y in the first equation and substituting into the second one gave the following equation in terms of BD:  2 2 BD BD 2BD 1  1  0. 4 2 This is a complex function involving both a quadratic function as well as a square root

function of BD. In order to solve this equation, the easiest way as shown in Figure 16 was to use my dynamic geometry software (Sketchpad) to quickly graph the function and find the solution for x = BD = 2.20 (rounded off to 2 decimals). From there one could easily use Pythagoras to determine BE, and use the trigonometric ratios to find all the angles, giving, for example, angle BAD = 112.28°. So as expected, this golden kite is slightly ‘fatter’ and more evenly proportionate than the previous two cases. One could therefore argue that it might be visually more pleasing also. golden ratio of sides or diagonals in the definition, substantially simplifies the deductive structure.golden This ratio illustrates of sides orthe diagonals important in educationalthe definition, point substantially that, generally, simplifies we thechoose deductive our mathematicalstructure. definitionsThis illustrates for convenienceforthe ‘convenience’ important andis educational the oneease byof which the point criteriathe otherthat, for generally, ‘convenience’A fourth we way choose to defineis the our and conceptualize a golden ease mathematicalby which the othedefinitionsr properties propertiesfor convenience can can be be derived derived and from fromone it. of it. the A fourthcriteria way for isoscelesto‘convenience’ define trapezium and conceptualize couldis the be to start aagain golden with isosceles trapezium could be to start a golden triangle ABX, but this time to translate ease by which the other properties can be derived from againit. with a goldenit with thetriangle vector BXABX along, but its this‘base’ time to produce to translate it with the vector BX along its a golden isosceles trapezium ABCD as shown in ‘base’ to produceFigure a golden11. In this isosceles case, since trapezium the figure is ABCD made as shown in Figure 11. In this case, since the figureup is of made3 congruent up of golden 3 congruent triangles, itgolden follows thattriangles, it follows that AB/AD = phi, AB/AD = phi, and BC = 2AD (and therefore its FigureFigure 13: Fir13. stFirst case case of of golden golden kite kite and BC = 2ADdiagonals (and therefore also divide its each diagonals other in the als ratioo divide each other in the ratio 2 to 1). 2 to 1). Though one could maybe argue that the first case of a golden isosceles trapezium A fourth way to define and conceptualize Thougha golden one isosceles could maybe trapezium argue that thecould first be case to start in Figure 8 is oftoo a golden‘broad’ isosceles and trapeziumthe one in FigureFigure 8 is11 too is too ‘tall’ to be visually appealing, again with a golden triangle ABX, but this time to translate it with the vector BX along its ‘broad’ and the one in Figure 11 is too ‘tall’ to be Figure 10. Third golden isosceles trapeziumthere is little visually appealing, different there betwe is littleen visually the differentone in Figure 10 and the second case in Figure‘base’ 10: toThird produce golden a golden isosceles isosceles trapezium trapezium ABCD as shown in Figure 11. In this case, Figure 8. However,between all the four one incases Figure or 10 types and the have second interesting case mathematical properties, and AnotherFiguresince completelythe 10: figure Third different isgolden made way isoscelesup to ofdefine 3 congruent trapeziumand in golden Figure 8. triangles, However, allit fourfollows cases orthat types AB have/AD = phi, Another completely different way to define and conceptualizedeserve to be a known. interestinggolden mathematicalisosceles properties, and deserve conceptualizeand BC = a2 goldenAD (and isosceles therefore trapezium its isdiagonals to also divide each other in the ratio 2 to 1). Another completely differentagain use way a golden to triangle.define As andshown conceptualize in Figure toa be golden known. isosceles Figure 14. Second case of golden kite Figure 14: Second case of golden kite trapezium is to again use a golden10, by reflectingtriangle.Though a Asgolden one shown trianglecould in ABCmaybe Figure in the argue 10, by that reflecting the first acase golden of a golden isosceles trapezium trapezium is to again use aperpendicular golden triangle. bisector ofAs one shown of its ‘legs’ in Figure BC, 10, by reflecting a golden triangle ABC in the perpendicular bisector of one of its ‘legs’ BC, produces a ‘golden ConstructivelyAnother way Definingmight be aagain ‘Golden to defineKite’ the pairs one,of angles and is intherefore the golden perhaps kite a littleto be more in the visually producesin Figure a ‘golden 8 is isosceles too ‘broad’ trapezium’ and where the onethe in Figure 11 is too ‘tall’ to be visually appealing, triangle ABC in the perpendicular bisector of one of its ‘legs’ BC, produces a ‘golden pleasing. This observation, of course, also relates isosceles trapezium’ where theratio ratioBC/AB BCis phi,/AB and is the phi, acute and ‘base’ the angle acute is ‘base’ angle is 72°. Againgolden there ratio are severalas shown possible in waysFigure in which14. Determining x from this geometric progression, there is little visually different between the one in Figure 10 and the second case in to constructively define the concept of a ‘golden to the ratio of the diagonals, which in the first isosceles trapezium’ where72°. theMoreover, ratio sinceBC /angleAB isBAD phi, = 108° and and the angle acute ‘base’ angle is 72°. 360 kite’.rounded An easy off way to oftwo constructing decimals, (andgives: defining) x  case 45 is .2.4084. (roundedOf special off interest to 2 decimals) is that thewhile in Moreover, since angle BAD =ADB 108° = 36°, and it follows angle that ADC angle =ABD 36°, is also it follows36°. that angle ABD is 2 Figure 8. However, all four cases or types have interesting mathematical properties, and 1 2the case in Figure 14, it is 1.84 (rounded off to Moreover, since angle BADHence, = 108°AD = ABand (= DCangle), and ADC therefore = 36°, the two it follows that angle ABD is one might be to again start with a golden triangle 2 decimals), and hence the latter is closer to the also 36°. Hence, AD = AB (=parallel deserveDC ),sides and to are be thereforealso known. in the golden the tworatio, andparallel as sides are also in the andangles construct at B anand equilateral D work triangleout to beon preciselyits base as equal to 120°. This golden kite looks a little golden ratio phi. also 36°. Hence, AD = ABwith (= the DC preceding), and case, therefore the diagonals the thereforetwo parallel sides are also in the shown in Figure 13. Since AB/BD = phi, it follows golden ratio, and as with the preceding case, the diagonals therefore also divide each ‘fatter’ than the precedingBD BC convex one, and is therefore perhaps a little more visually also divide each other into the golden ratio. Of immediately that since =  by construction, To define a golden kite that is hopefully even golden ratio, and as with the preceding case, the diagonals therefore also divideFigure each 12: Golden isosceles trapezia of type 3 AB to BC is also in the golden ratio. Notice that other into the golden ratio. Ofinterest interest also, alsois to note is tothat note the diagonals that the AC diagonals and AC and DB each pleasing. This observation, of course, also relatesmore to the visually ratio appealing of the diagonals, than the previous which intwo, I DB each respectively bisect the ‘base’ angles2 at C the same construction applies to the concave case, other into the golden ratio. Of interest also is to note that the diagonals AC and DB each the first case is 2.40 (rounded off to 2 decimals) nextwhile thought in the ofcase defining in Figure a ‘golden 14, it kite’is 1.84 as shown and B. 2 but is probably not as ‘visually pleasing’ as the 3 respectively bisect the ‘base’ angles at 2B and C. One more argument towards perhaps slightly favoring the golden isosceles trapezium, in Figure 15, namely, as a (convex ) kite with respectively bisect the ‘base’ angles at B and C. convex(rounded case. off to 2 decimals), and hence the latter isboth closer its sides to the and golden diagonals ratio in phi. the golden ratio. defined and constructed in Figure 9, might be that it appears in both the regular convex Another way might be again to define the pairs To define a golden kite that is hopefully even more visually appealing than the pentagon as well as the regular star pentagon as illustrated in Figure 12. of angles in the golden kite to be in the golden previous two, I next thought of defining a ‘golden kite’ as shown in Figure 15, namely, as ratio as shown in Figure 14. Determining x from 3 Constructively Defining a ‘Golden Kite’ thisa (convexgeometric) progression,kite with both rounded its sides off andto two diagonals in the golden ratio. Figure 12. Golden isosceles trapezia of type 3 decimals, gives: FigureAgain 12: there Golden are isoscelesseveral possible trapezia ways of type in which3 to constructively define the concept of a One more argument towards perhaps slightly 360° x = = 45.84° 2 One more argument towards‘golden perhaps kite’. slightly Anfavoring easy favorin the way golden gof the constructingisosceles golden trapezium, isosceles (and defined defining) trapezium, one might be to again start with 1+φ + 2φ and constructed in Figure 10, might be that it defined and constructed in Figurea golden 9, trianglemight be and that construct it appears an inequilateral both the regulartriangle convexon its base as shown in Figure 13. appears in both the regular convex pentagon as Of special interest is that the angles atFi Bgure and 15:D Third case: Golden kite with sides and diagonals in the golden ratio FigureFigure 11: 11:A fourth A fourth golden golden isosceles isosceles trapezium trapezium well as the regular star pentagon as illustrated in Figure 15. Third case: Golden kite with sides and pentagonFigure 11. asA fourth well golden as the isosceles regular trapeziumSince star AB pentagon/BD = phi, as illustratedit follows immediatelyin Figure 12. that since BD = BC by construction, ABwork to out BC to be precisely equal to 120°. This golden diagonals in the golden ratio 1 Figure 12. �� 1 kite3 looks a little ‘fatter’ than theThough preceding one convex can drag a dynamically constructed kite in dynamic geometry with sides is also in the golden ratio. Notice that the same construction applies to the concave For case, the sake of brevity we shall disregard the concave case here. Constructively Defining a ‘Golden Kite’ constructed in the golden ratio so that its diagonals are approximately also in the golden 2 In De Villiers (2009, p. 154-155; 207) a general isoscelesbut trapezium is probably with three adjnotacent as sides ‘visually equal is called apleasing’ trilateral trapezium, as andthe the convex property case. thatAgain a pair of adjacent,there congruentare several angles are bisectedpossible by the diagonalsways is inalso mentioned.which Alsoto see:constructively http://dynamicmathematicslearning.com/quad- define the concept of a ratio, making an accurate construction required the calculation of one of the angles. At tree-new-web.html 3 For the sake of brevity we shall disregard the concave case here. 2 2 ‘golden kite’. An easy way of constructing (and defining) one might be to again start with first I again tried to use the cosine rule, since it had proved effective in the case of one In De In Villiers De Villiers (2009, (2009, p. 154 p.- 155;154- 155;207) 207) a general a general isosceles isosceles trapezium trapezium with with three three adjacent adjacent sides sides equal equal is is calledcalled a trila ateral trila teraltrapezium trapezium, and76, theand property atheAt golden Right property Angles that triangle thata| Vol.pair a 6, pair No.ofand adjacent,2, of August constructadjacent, 2017 congruent congruent an equilateral angles angles are are trianglebisected bisected byon by the its the base as shown in Figure 13. golden parallelogram as well asAt Rightone Anglesisosceles | Vol. 6,trapezium No. 2, August case, 2017 but77 with no success. diagonalsdiagonals is also is mentioned.also mentioned. Also Also see: see:http://dynamicmathematicslearning.com/quad http://dynamicmathematicslearning.com/quad-tree-tree-new-new-web.html-web.html Eventually switching strategies, and assuming AB = 1, applying the theorem of Since AB/BD = phi, it follows immediately that since BD = BC by construction, AB to BC Pythagoras to the right triangles ABE and ADE gave the following: is also in the golden ratio. Notice that the same construction applies to the concave case, 2 At Right Angles | Vol. 6, No. 2, August 2017 PB 2 BD y  1 but is probably not as ‘visually pleasing’ as the convex case. 4 2

since its midpoint rhombus would then be a golden rhombus4 (with diagonals in golden Figure 16: Solving for BD by graphing 4 ‘golden kites’, which are illustrated in Figure 19. As can be seen, they can be obtained Figure 17: Fifth case: Golden isosceles trapezia via midsegments in golden ratio In De Villiers (2009, p. 154 In- 155;De Villiers 207) (2009, , a general p. 154-155; kite 207) with , a threegeneral adjacentkite with three angles adjacent equal angles is calledequal is calleda a triangular kite, and the property that a pair of Figure 16. Solving for BD by graphing ratio). However,Figure 17. Fifth in general case: Golden such isosceles an isosceles trapezia trapezium istriangular dynamic kite and, and can the change propertyadjacent, shape, that congruent a pair of sides adjacent, are bisected congruent by the tangent sides points are of bisected the incircle by is thealso mentioned.tangent points The Penrose kite in Figure 19 is also an example of a from a rhombus with angles of 72° and 108° by dividing the long diagonal of the In addition, the midpoint rectangle of the third golden kite in Figure 15, sincevia midsegmentsits in golden ratio of the incircle is also mentioned.st triangular The kite. Penrose Also see: kite http://dynamicmathematicslearning.com/quad-tree-new-web.html in Figure 19 is also an example of a triangular kite. Also On that note, jumping back to the andprevious we need section, to add this a furtherreminded property me that to afix fifth its wayshape. in Forsee: example, http://dynamicmathemat in the 1 caseicslearning.com/quad shown -tree-new-web.html diagonals are in the golden ratio, is a golden rectangle. rhombus in the ratio of phi so that the ‘symmetrical’ diagonal of the Penrose kite is in the nd 78 At Right Angleswhich | Vol. 6,we No. could 2, August define 2017 a golden isoscelesin Figure trapezoid 17 we could might impose be to define the condition it as an thatisosceles BC/AD = phi, or as in the 2 case, we At Right Angles | Vol. 6, No. 2, August 2017 79 ratio phi to the ‘symmetrical’ diagonal of the dart. It is left to the reader to verify that trapezium with its mid-segments KMcan andhave LN AB in = the AD golden = DC (soratio the as base shown angles in Figureat B and 17, C would respectively be bisected by from this construction it follows that both the Penrose kite and dart have their sides in the since its midpoint rhombus would thethen diagonals be a golden DB rhombusand AC). (withAs can diagonals be seen, in itgolden is very difficult to visually distinguish ratio of phi. Moreover, the Penrose kites and darts can be used to tile the plane non- ratio). However, in general such anbetween isosceles these trapezium two cases is dynamic since the and angles can change only differshape, by a few degrees (as can be easily At Right Angles | Vol. 6, No. 2, August 2017 PB periodically, and the ratio of the number of kites to darts tends towards phi as the number and we need to add a further propertyverified to fix by its calculation shape. For byexample, the reader). in the Also 1st case note shown that for the construction in Figure 17, as of tiles increase (Darvas, 2007: 204). Of additional interest, is that the ‘fat’ rhombus Figure 17: Fifth case: Goldenin isoscelesFigure 17 trapezia we could via midsegments impose the conditionin golden ratiothat BC/AD = phi, or as in the 2nd case, we formed by the Penrose kite and dart as shown in Figure 19, also non-periodically tiles can have AB = AD = DC (so the base angles at B and C would respectively be bisected by On that note, jumping back to the previous section, this reminded me that a fifth way in with the ‘thin’ rhombus given earlier by the second golden rhombus in Figure 7, and the the diagonals DB and AC). As can be seen, it is very difficult to visually distinguish which we could define a golden isosceles trapezoid might be to define it as an isosceles ratio of the number of ‘fat’ rhombi to ‘thin’ rhombi similarly tends towards phi as the trapezium with its mid-segmentsbetween KM and theseLN in two the casesgolden since ratio theas shown angles in only Figure differ 17, by a few degrees (as can be easily number of tiles increase (Darvas, 2007: 202). The interested reader will find various since its midpoint rhombus wouldverified then beby acalculation golden rhombus by the (with reader). diagonals Also notein golden that for the construction in Figure 17, as websites on the Internet giving examples of Penrose tiles of kites and darts as well as of ratio). However, in general such an isosceles trapezium is dynamic and can change shape, the mentioned rhombi. and we need to add a further property to fix its shape. For example, in the 1st case shown in Figure 17 we could impose the condition that BC/AD = phi, or as in the 2nd case, we Constructively Defining Other ‘Golden Quadrilaterals’ can have AB = AD = DC (so the base angles at B and C would respectively be bisected by This investigation has already become longer than I’d initially anticipated, and it is time the diagonals DB and AC). As can be seen, it is very difficult to visually distinguish to finish it off before I start boring the reader. Moreover, my main objective of showing between these two cases since the angles only differ by a few degrees (as can be easily constructive defining in action has hopefully been achieved by now. verified by calculation by the reader). Also note that for the construction in Figure 17, as However, I’d like to point out that there are several other types of quadrilaterals for which one can similarly explore ways to define ‘golden quadrilaterals’, e.g., cyclic Though one can drag a dynamically constructed quickly graph the function and find the solution cases since the angles only differ by a few degrees decimals), and since it is further from the golden kite in dynamic geometry with sides constructed for x = BD = 2.20 (rounded off to 2 decimals). (as can be easily verified by calculation by the ratio, explains the elongated, thinner shape in in the golden ratio so that its diagonals are From there one could easily use Pythagoras to reader). Also note that for the construction in comparison with the golden kites in Figures 14 approximately also in the golden ratio, making determine BE, and use the trigonometric ratios Figure 17, as we’ve already seen earlier, AD to AB and 15. a b + c a + b + c we’ve already seen earlier, AD to AB will be in the golden ratio, if and only if, isosceles anAF accurate= c construction− = required− the calculation= s a; to find all the angles, giving, for example, angle will be in the golden ratio, if and only if, isosceles of one of− the angles.2 At first I 2again tried to− use BAD = 112.28°. So as expected, this golden kite trapezium ABCD is a goldentrapezium rectangle. ABCD is a golden rectangle. Last, but not least, one can also choose to define the cosine rule, since it had proved effective in is slightly ‘fatter’ and more evenly proportionate the famous Penrose kite and dart as ‘golden kites’, a + b + c a + b c theCE case= b of one− golden parallelogram= − as= wells asc , than the previous two cases. One could therefore which are illustrated in Figure 19. As can be seen, − a b + c a + b + c − oneAF isosceles= c trapezium− 2 = case,− but2 with no= ssuccess.a; argue that it might be visually more pleasing also. they can be obtained from a rhombus with angles − 2 2 − of 72° and 108° by dividing the long diagonal Eventually switchinga b + strategies,c a + band cassuming AB = CD1, applying= a the− theorem= of Pythagoras− = sto thec; In addition, the midpoint rectangle of the third of the rhombus in the ratio of phi so that the − a +2b + c a +2b c − rightCE = trianglesb − ABE and ADE= gave −the following:= s c, golden kite in Figure 15, since its diagonals are in ‘symmetrical’ diagonal of the Penrose kite is in − 2 2 − the golden ratio, is a golden rectangle. the ratio phi to the ‘symmetrical’ diagonal of the BD 2 dart. It is left to the reader to verify that from a by+2 +c a +=b 1 c On that note, jumping back to the previous CD = a − =4φ 2 − = s c; this construction it follows that both the Penrose − 2 2 − section, this reminded me that a fifth way in kite and dart have their sides in the ratio of phi. BD 2 which we could define a golden isosceles trapezoid +(BD y) 2 = φ 2. Moreover, the Penrose kites and darts can be used 2 − 2 might be to define it as an isosceles trapezium 4φ 2 BD to tile the plane non-periodically, and the ratio of y + = 1 with its mid-segments KM and LN in the golden 4φ 2 the number of kites to darts tends towards phi as Solving for y in the first equation2 and substituting ratio as shown in Figure 17, since its midpoint 2 2 BD the number of tiles increase (Darvas, 2007: 204). intoBD the secondBD2BD one1 gave the 2followingφ +2 1 equation= 0. rhombus would then be a golden rhombus (with − +(BD− y2) −= φ . Of additional interest, is that the ‘fat’ rhombus 2 √ −4φ in terms of 4BDφ : diagonals in golden ratio). However, in general, formed by the Penrose kite and dart as shown such an isosceles trapezium is dynamic and can U = U + r (t t ), in Figure 19, also non-periodically tiles with 0 U 2 0 change shape, and we need to add a further 2 BD − the ‘thin’ rhombus given earlier by the second BD 2BD 1 φ + 1 = 0. property to fix its shape. For example, in the 1st − √ − 4φ 2 − Figure 18: Constructing golden kite tangent to circumcircle of KLMNgolden rhombus in Figure 7, and the ratio of the Flea case shown in Figure 17 we could impose the Figure 18. Constructing golden kite tangent to lim = 0. nd circumcircle of KLMN number of ‘fat’ rhombi to ‘thin’ rhombi similarly This is a complexh function0 h involving both a condition that BC/AD = phi, or as in the 2 case, Since all isosceles trapezia are cyclic (and all kites are circumscribed), another way to U = →U0 + rU(t t0), tends towards phi as the number of tiles increase quadratic function as well as a− square root we can have AB = AD = DC (so the base angles conceptualize and constructivelySince all isosceles define trapezia a ‘golde are cyclicn kite’ (and would all kites be to also(Darvas, construct 2007: the 202). The interested reader will function of BDΔ.f To= hfsolve′(x )+this Fleaequation,, the at B and C would respectivelyFigure be 16: bisected Solving by thefor BD by graphing are circumscribed), another way to conceptualize Flea find various websites on the Internet giving easiest way as shownlim in Figure= 0 .16 was to use diagonals DB and AC). As can be seen, it is very ‘dual’ of each of the goldenand constructively isosceles trapezia define a already‘golden discussed.kite’ would For example,examples consider of Penrose tiles of kites and darts as well my dynamic geometryh 0 hsoftware (Sketchpad) to In difficultaddition, tothe visually midpoint distinguish rectangle between of thesethe thirdtwo golden kite in Figure 15, since its → the golden isosceles trapeziumbe to also KLMNconstruct defined the ‘dual’ in Figure of each 10, of theand its circumcircleas of the as mentioned shown rhombi. diagonals are in the golden ratio, is a golden rectangle. golden isosceles trapezia already discussed. For Δf = hf ′(x)+Flea, in Figure 18. As was example,the case considerfor the gothelden golden rectangle, isosceles wetrapezium can now similarly construct perpendiculars to the radiiKLMN at eachdefined of thein Figurevertices 10, to and produce its circumcircle a corresponding dual ‘golden as shown in Figure 18. As was the case for the Figure 16: Solving for BD by graphing kite’ ABCD. It is nowgolden left to rectangle, the reader we canto verify now similarly that CBD construct is a golden triangle (hence BC/BD = phi) and angleperpendiculars ABC = angle to theBAD radii = angleat each ADC of the = vertices 108°. In addition ABCD has In addition, the midpoint rectangle of the third golden kite in Figure 15, since its to produce a corresponding dual ‘golden kite’ the dual property (to theABCD angle. It bisection is now left of to two the anglesreader toby verify diagonals that in KLMN) of K and N diagonals are in the golden ratio, is a golden rectangle. being respective midpointsCBD is of a goldenAB and triangle AD4 .(hence The readerBC/BD may= phi) also wish to verify that Figure 17: Fifth case: Golden isosceles trapezia via midsegments in goldenand ratio angle ABC = angle BAD = angle ADC = 108°. AC/BD = 1.90 (roundedIn additionoff to 2 ABCDdecimals), has the and dual since property it is further(to the from the golden ratio, On that note, jumping back to the previous section, this reminded me that a fifth way in explains the elongated,angle thinner bisection shape of intwo comparison angles by diagonals with the in golden kites in Figures 14 KLMN) of K and N being respective midpoints which we could define a golden isosceles trapezoid might be to define it as an isosceles FigureFigure 19: 19. Penrose Penrose kite kite and and dart dart and 15. of AB and AD . The reader may also wish to trapezium with its mid-segments KM and LN in the golden ratio as shown in Figureverify that 17, AC/BD 1 = 1.90 (rounded off to 2 Last, but not least, one can also choose to define the famous Penrose kite and dart as since its midpoint rhombus would then be a golden rhombus4 (with diagonals in golden Figure 16: Solving for BD by graphing 4 ‘golden kites’, which are illustrated in Figure 19. As can be seen, they can be obtained Figure 17: Fifth case: Golden isosceles trapezia via midsegments in golden ratio In De Villiers (2009, p. 154 In- 155;De Villiers 207) (2009, , a general p. 154-155; kite 207) with , a threegeneral adjacentkite with three angles adjacent equal angles is calledequal is calleda a triangular kite, and the property that a pair of Figure 16. Solving for BD by graphing ratio). However,Figure 17. Fifth in general case: Golden such isosceles an isosceles trapezia trapezium istriangular dynamic kite and, and can the change propertyadjacent, shape, that congruent a pair of sides adjacent, are bisected congruent by the tangent sides points are of bisected the incircle by is thealso mentioned.tangent points The Penrose kite in Figure 19 is also an example of a from a rhombus with angles of 72° and 108° by dividing the long diagonal of the In addition, the midpoint rectangle of the third golden kite in Figure 15, sincevia midsegmentsits in golden ratio of the incircle is also mentioned.st triangular The kite. Penrose Also see: kite http://dynamicmathematicslearning.com/quad-tree-new-web.html in Figure 19 is also an example of a triangular kite. Also On that note, jumping back to the andprevious we need section, to add this a furtherreminded property me that to afix fifth its wayshape. in Forsee: example, http://dynamicmathemat in the 1 caseicslearning.com/quad shown -tree-new-web.html diagonals are in the golden ratio, is a golden rectangle. rhombus in the ratio of phi so that the ‘symmetrical’ diagonal of the Penrose kite is in the nd 78 At Right Angleswhich | Vol. 6,we No. could 2, August define 2017 a golden isoscelesin Figure trapezoid 17 we could might impose be to define the condition it as an thatisosceles BC/AD = phi, or as in the 2 case, we At Right Angles | Vol. 6, No. 2, August 2017 79 ratio phi to the ‘symmetrical’ diagonal of the dart. It is left to the reader to verify that trapezium with its mid-segments KMcan andhave LN AB in = the AD golden = DC (soratio the as base shown angles in Figureat B and 17, C would respectively be bisected by from this construction it follows that both the Penrose kite and dart have their sides in the since its midpoint rhombus would thethen diagonals be a golden DB rhombusand AC). (withAs can diagonals be seen, in itgolden is very difficult to visually distinguish ratio of phi. Moreover, the Penrose kites and darts can be used to tile the plane non- ratio). However, in general such anbetween isosceles these trapezium two cases is dynamic since the and angles can change only differshape, by a few degrees (as can be easily At Right Angles | Vol. 6, No. 2, August 2017 PB periodically, and the ratio of the number of kites to darts tends towards phi as the number and we need to add a further propertyverified to fix by its calculation shape. For byexample, the reader). in the Also 1st case note shown that for the construction in Figure 17, as of tiles increase (Darvas, 2007: 204). Of additional interest, is that the ‘fat’ rhombus Figure 17: Fifth case: Goldenin isoscelesFigure 17 trapezia we could via midsegments impose the conditionin golden ratiothat BC/AD = phi, or as in the 2nd case, we formed by the Penrose kite and dart as shown in Figure 19, also non-periodically tiles can have AB = AD = DC (so the base angles at B and C would respectively be bisected by On that note, jumping back to the previous section, this reminded me that a fifth way in with the ‘thin’ rhombus given earlier by the second golden rhombus in Figure 7, and the the diagonals DB and AC). As can be seen, it is very difficult to visually distinguish which we could define a golden isosceles trapezoid might be to define it as an isosceles ratio of the number of ‘fat’ rhombi to ‘thin’ rhombi similarly tends towards phi as the trapezium with its mid-segmentsbetween KM and theseLN in two the casesgolden since ratio theas shown angles in only Figure differ 17, by a few degrees (as can be easily number of tiles increase (Darvas, 2007: 202). The interested reader will find various since its midpoint rhombus wouldverified then beby acalculation golden rhombus by the (with reader). diagonals Also notein golden that for the construction in Figure 17, as websites on the Internet giving examples of Penrose tiles of kites and darts as well as of ratio). However, in general such an isosceles trapezium is dynamic and can change shape, the mentioned rhombi. and we need to add a further property to fix its shape. For example, in the 1st case shown in Figure 17 we could impose the condition that BC/AD = phi, or as in the 2nd case, we Constructively Defining Other ‘Golden Quadrilaterals’ can have AB = AD = DC (so the base angles at B and C would respectively be bisected by This investigation has already become longer than I’d initially anticipated, and it is time the diagonals DB and AC). As can be seen, it is very difficult to visually distinguish to finish it off before I start boring the reader. Moreover, my main objective of showing between these two cases since the angles only differ by a few degrees (as can be easily constructive defining in action has hopefully been achieved by now. verified by calculation by the reader). Also note that for the construction in Figure 17, as However, I’d like to point out that there are several other types of quadrilaterals for which one can similarly explore ways to define ‘golden quadrilaterals’, e.g., cyclic Constructively Defining Other equi-angled, cyclic hexagon that all the pairs of compared in terms of the number of properties, investigation has hopefully also contributed a ‘Golden Quadrilaterals’ adjacent sides as shown in Figure 19 are in the ease of construction or of proof, and, in this little bit to demystifying where definitions come This investigation has already become longer than golden ratio; i.e., a ‘golden (cyclic) hexagon’. It particular case in relation to the golden ratio, from, and that they don’t just pop out of the I’d initially anticipated, and it is time to finish it is left to the reader to verify that if FA/AB = phi, perhaps also of visual appeal. Moreover, it was air into a mathematician’s mind or suddenly 7 off before I start boring the reader. Moreover, my then AL/LM = phi , etc. In other words, the main shown how some definitions of the same object magically appear in print in a school textbook. main objective of showing constructive defining diagonals divide each other into the golden ratio. might be more convenient than others in terms of Referencesthe deductive derivation of other properties not In a classroom context, if a teacher were to ask in action has hopefully been achieved by now. The observant reader would also note that ABEF, students to suggest various possible definitions 1.containedDe Villiers, in M.the (2009). definition.Some Adventures in Euclidean Geometry. Lulu Press. However, I’d like to point out that there are ABCD and CDEF, are all three golden trapezia for golden quadrilaterals or golden hexagons several other types of quadrilaterals for which of the type constructed and defined in the first 2.TheErnest, process P. (1991). of constructiveThe Philosophy defining of Mathematics also generally Education , London,of different Falmer Press. types, it is likely that they would case in Figure 8. Moreover, ALNF, ABCF, etc., applies to the definition and exploration of propose several of the examples discussed here, one can similarly explore ways to define ‘golden 3. Freudenthal, H. (1973). Mathematics as an Educational Task. D. Reidel, Dordrecht, Holland. quadrilaterals’, e.g., cyclic quadrilaterals, are golden trapezia of the second type constructed different axiom systems in pure, mathematical and perhaps even a few not explored here. circumscribed quadrilaterals, trapeziums5, and defined in Figure 8. 4.researchGrossman, where P. et quite al. (2009). often Do existing People Preferaxiom Irrational Ratios? AInvolving New Look students at the Golden in an Section. activity Student like researchthis would conducted in 2008/2009 in the Dept. of Applied Computer Science, University of Bamberg. Accessed on 23 Oct 2016 at: 5 systems are used as starting blocks which are then not only more realistically simulate actual quadrilaterals, circumscribedbi-centric quadrilaterals,quadrilaterals, orthodiagonal trapeziums , bi-centric quadrilaterals, https://www.academia.edu/3704076/The_Golden_Ratio quadrilaterals, equidiagonal quadrilaterals, etc. modified, adapted, generalized, etc., to create mathematical research, but also provide students orthodiagonal quadrilaterals, equidiagonal quadrilaterals, etc. 5.andLoeb, explore A.L. new & Varney, mathematical W. (1992). Doestheories. the Golden So this Spiral Exist,with and If a Not, more Where personal is its Center? sense of In Hargittai,ownership I. & over Pickover, the C.A. little(1992). episodeSpiral encapsulates Similarity. Singapore: at an elementary World Scientific, level pp. 47-63.mathematical content instead of being seen as 6.someSerra, of M.the (2008, main Editionresearch 4). methodologiesDiscovering Geometry: used An Investigativesomething Approach. that Emeryville: is only Key the Curriculum privilege Press.of some select by research mathematicians. In that sense, this mathematically endowed individuals. 7. Stieger, S. & Swami, V. (2015). Time to let go? No automatic aesthetic preference for the golden ratio in art pictures. Psychology of Aesthetics, Creativity, and the Arts, Vol 9(1), Feb, 91-100. http://dx.doi.org/10.1037/a0038506

8. Walser, H. (2001). The Golden Section. Washington, DC: The Mathematical Association of America. Reference:

Endnotes1. Darvas, G. (2001). Symmetry. Basel: Birkhäuser Verlag. Figure 21: Cutting off two rhombi and a golden trapezium Figure 21. Cutting off two rhombi and a golden 2. De Villiers, M. (2009). Some Adventures in Euclidean Geometry. Lulu Press. 1. This is not the common textbook definition. (The usual definition is: A parallelogram is a four-sided figure for whichbothpairsof trapezium 3. De Villiers, M. (2011). Equi-angled cyclic and equilateral circumscribed polygons. The Mathematical Gazette, 95(532), By cutting off two rhombi and a golden isosceles trapezium as shown in Figure 21, we oppositeMarch, pp. sides 102-106. are parallel Accessed to each 16 October other.) I2016 want at: to http://dynamicmathematiclearning.com/equi-anglecyclicpoly.pdf emphasize that concepts can be defined differently and often more powerfully in Figure 20: A golden hexagon with adjacent sides in golden ratio terms of symmetry. As argued in De Villiers (2011), it is more convenient defining quadrilaterals in terms of symmetry than the By cutting off two rhombi and a golden isosceles 4. De Villiers, M. (2016). Enrichment for the Gifted: Generalizing some Geometrical Theorems & Objects.Learning Figure 20. A golden hexagonalso withobtain adjacent a similar golden cyclic hexagon. Lastly, it is also left to the reader to consider, standard textbook definitions. Reference: De Villiers, M. (2011). Simply Symmetric. Mathematics Teaching, May 2011, p34–36. trapezium as shown in Figure 21, we also obtain a and Teaching Mathematics, December 2016. Accessed 16 October 2016 at: http://dynamicmathematicslearning.com/ Constructively Defining a ‘Golden sidesCyclic in golden Hexagon’ ratio define and investigate an analogoussimilar golden dual cyclic of a hexagon. golden cyclic Lastly, hexagon. it is also left 2. TheICME13-TSG4-generalization.pdf Golden Ratio can be defined in different ways. The simplest one is: it is that positivenumber x for which x = 1 + 1/x; 2 √ Before closing, I’d like to briefly tease the reader with considering investigatingto the reader defining to consider, define and investigate 5. equivalently,De Villiers, M. that (2017). positive An numberexample xoffor constructive which x defining:= x + 1. TheFrom a definitiongolden rectangle implies to golden that x =( quadrilaterals5 + 1)/2, whose& value is approximately 1.618034. A rectangle whose length : width ratio is x : 1 is known as a golden rectangle. It has the feature that when Constructively Defining aConcluding Remarks an analogous dual of a golden cyclic hexagon. beyond: Part 1. At Right Angles. Volume 6, No. 1, (March), pp. 64-69. hexagonal ‘golden’ analogues for at least some of the golden quadrilaterals discussed we remove the largest possible square from it (a 1 by 1 square), the rectangle that remains is again a golden rectangle. ‘Golden Cyclic Hexagon’ Though most of the mathematical results discussed here are not novel, it is hoped that this 6. Olive, J. (Undated). Construction and Investigation of Golden Trapezoids. University of Georgia, Athens; 3. TheUSA: termClassroom Golden Notes. Rectangle Accessed 23 has October by now 2016 a standard at: http://math.coe.uga.edu/olive/EMAT8990FYDS08/ meaning. However, terms like Golden Rhombus, Golden Parallelogram, Golden here. For example, theBefore analogous closing, I’dequivalent like to briefly of a teaserectangle the reader is an equi-angled, cyclic little investigation has toConcluding some extent Remarks shown the productive process of constructive TrapeziumGoldenTrapezoids.doc and Golden Kite have been defined in slightly different ways by different authors. 6 hexagon as pointed without consideringin De Villiers defining (2011; hexagonal 2016). ‘golden’ Hence, one possibleThough waymost ofto the mathematical results analogues for at least some ofdefining the golden by illustrating how new mathematical objects can be defined and constructed construct a hexagonal analogue for the golden rectangle is to impose thediscussed condition here on are an not novel, it is hoped that this quadrilaterals discussed here.from For example,familiar thedefinitions littleof known investigation objects. has In to the some pr ocess,extent shownseveral different possibilities analogous equivalent of a rectangle is an equi- MICHAEL DE VILLIERS has worked as researcher, mathematics and science teacher at institutions across equi-angled, cyclic hexagon that all1 the pairs of adjacent sides as shownthe in Figureproductive 19 areprocess of constructive defining MICHAEL DE VILLIERS 6 may be explored and compared in terms of the number of properties, ease of construction the world. Since 1991 he has workedbeen part as researcher,of the University mathematics of Durban-Westville and science teacher (now at institutions University across of KwaZulu- the angled, cyclic hexagon as pointed out in De world. He retired from the University of KwaZulu-Natal at the end of January 2016. He has subsequently been in the golden ratio; i.e., a ‘golden (cyclic) hexagon’. It is left to the readerby to illustrating verify that how if new mathematical objects Natal). He was editor of Pythagoras, the research journal of the Association of Mathematics Education of South Villiers (2011; 2016). Hence,or oneof proof,possible and way, in this particularcan be defined case, andperh constructedaps also of from visual familiar appeal. Moreover, it was appointed Professor Extraordinaire in Mathematics Education at the University of Stellenbosch. He was editor of to construct a7 hexagonal analogue for the golden Africa,Pythagoras and, the has research been vice-chairjournal of theof Associationthe SA Mathematics of Mathematics Olympiad Education since of 1997.South Africa,His main and wasresearch vice-chair interests of are FA/AB = phi, then AL/LM = phi , etc. In other words, the main diagonalsdefinitions divide ofeach known objects. In the process, Geometry, Proof, Applications and Modeling, Problem Solving, and the History of Mathematics. His home rectangle is to impose the conditionshown how on an some definitions of the same object might be more convenient than others in the SA Mathematics Olympiad from 1997 -2015, but is still involved. His main research interests are Geometry, other into the golden ratio. several different possibilities may be explored and pageProof, is Applications http://dynamicmathematics-learning.com/homepage4.html and Modeling, Problem Solving, and the History of. Mathematics.He maintains His a homeweb pagepage is forhttp:// dynamic terms of the deductive derivation of other properties not contained in the definition. geometrydynamicmathematics-learning.com/homepage4.html sketches at http://dynamic-mathematicslearning.com/JavaGSPLinks.htm. He maintains a web page for dynamic. He geometrymay be sketchescontacted on The observant reader would also note that ABEF, ABCD and CDEF, are all three [email protected] http://dynamic-mathematicslearning.com/JavaGSPLinks.htm. . He may be contacted on [email protected]. 5 The process of constructive defining also generally applies to the definition and golden trapezia of the Olive type (undated), constructed for example, constructively and defined defines twoin different,the firs interestingt case types in of goldenFigure trapezoids/trapeziums. 8. 6 This type of hexagon is also called a semi-regularexploration angle-hexagon of different in the referenced axiom papers. systems in pure, mathematical research where quite often Moreover, ALNF, ABCF7 It was, withetc. surprised, are goldeninterest that trapezia in October 2016,of theI came second upon Odom’s type construction constructed at: http://demonstrations.wolfram.com/ and HexagonsAndTheGoldenRatio/, whichexisting is the converse axiom of this result.systems With reference are used to the figure,as starting Odom’s construction blocks involves which extending are thenthe sides modified, adapted, defined in Figure 8. of the equilateral triangle LMN to construct three equilateral triangles ABL, CDM and EFN. If the extension is proportional to the golden ratio, then the outer vertices of these three trianglesgeneralized, determine a etc.(cyclic,, toequi-angled) create hexagon and explore with adjacent new sides inmathematical the golden ratio. theories. So this little episode encapsulates at an elementary level some of the main research methodologies used by 80 At Right Angles | Vol. 6, No. 2, August 2017 At Right Angles | Vol. 6, No. 2, August 2017 81 5 research mathematicians. In that sense, this investigation has hopefully also contributed a Olive (undated), for example, constructively defines two different, interesting types of golden trapezoids/trapeziums. little bit to demystifying where definitions come from, and that they don’t just pop out of 6 This type of hexagon is also called a semi-regular angle-hexagon in the referenced papers. 7 the air into a mathematician’s mind or suddenly magically appear in print in a school It was with surprised interest that in October 2016, I came upon Odom’s construction at: At Right Angles | Vol. 6, No. 2, August 2017 PB http://demonstrations.wolfram.com/HexagonsAndTheGoldenRatio/textbook., which is the converse of this result. With reference to the figure, Odom’s construction involves extending the sides of the equilateral triangle LMN to construct three equilateral triangles ABL, CDM and EFN. InIf thea extensionclassroom is proportional context, ifto thea teacher were to ask students to suggest various golden ratio, then the outer vertices of these three triangles determine a (cyclic, equi-angled) hexagon with At Right Angles | Vol. 6, No. 1, March 2017 69 adjacent sides in the golden ratio. possible definitions for golden quadrilaterals or golden hexagons of different types, it is Constructively Defining Other equi-angled, cyclic hexagon that all the pairs of compared in terms of the number of properties, investigation has hopefully also contributed a ‘Golden Quadrilaterals’ adjacent sides as shown in Figure 19 are in the ease of construction or of proof, and, in this little bit to demystifying where definitions come This investigation has already become longer than golden ratio; i.e., a ‘golden (cyclic) hexagon’. It particular case in relation to the golden ratio, from, and that they don’t just pop out of the I’d initially anticipated, and it is time to finish it is left to the reader to verify that if FA/AB = phi, perhaps also of visual appeal. Moreover, it was air into a mathematician’s mind or suddenly 7 off before I start boring the reader. Moreover, my then AL/LM = phi , etc. In other words, the main shown how some definitions of the same object magically appear in print in a school textbook. main objective of showing constructive defining diagonals divide each other into the golden ratio. might be more convenient than others in terms of Referencesthe deductive derivation of other properties not In a classroom context, if a teacher were to ask in action has hopefully been achieved by now. The observant reader would also note that ABEF, students to suggest various possible definitions 1.containedDe Villiers, in M.the (2009). definition.Some Adventures in Euclidean Geometry. Lulu Press. However, I’d like to point out that there are ABCD and CDEF, are all three golden trapezia for golden quadrilaterals or golden hexagons several other types of quadrilaterals for which of the type constructed and defined in the first 2.TheErnest, process P. (1991). of constructiveThe Philosophy defining of Mathematics also generally Education , London,of different Falmer Press. types, it is likely that they would case in Figure 8. Moreover, ALNF, ABCF, etc., applies to the definition and exploration of propose several of the examples discussed here, one can similarly explore ways to define ‘golden 3. Freudenthal, H. (1973). Mathematics as an Educational Task. D. Reidel, Dordrecht, Holland. quadrilaterals’, e.g., cyclic quadrilaterals, are golden trapezia of the second type constructed different axiom systems in pure, mathematical and perhaps even a few not explored here. circumscribed quadrilaterals, trapeziums5, and defined in Figure 8. 4.researchGrossman, where P. et quite al. (2009). often Do existing People Preferaxiom Irrational Ratios? AInvolving New Look students at the Golden in an Section. activity Student like researchthis would conducted in 2008/2009 in the Dept. of Applied Computer Science, University of Bamberg. Accessed on 23 Oct 2016 at: 5 systems are used as starting blocks which are then not only more realistically simulate actual quadrilaterals, circumscribedbi-centric quadrilaterals,quadrilaterals, orthodiagonal trapeziums , bi-centric quadrilaterals, https://www.academia.edu/3704076/The_Golden_Ratio quadrilaterals, equidiagonal quadrilaterals, etc. modified, adapted, generalized, etc., to create mathematical research, but also provide students orthodiagonal quadrilaterals, equidiagonal quadrilaterals, etc. 5.andLoeb, explore A.L. new & Varney, mathematical W. (1992). Doestheories. the Golden So this Spiral Exist,with and If a Not, more Where personal is its Center? sense of In Hargittai,ownership I. & over Pickover, the C.A. little(1992). episodeSpiral encapsulates Similarity. Singapore: at an elementary World Scientific, level pp. 47-63.mathematical content instead of being seen as 6.someSerra, of M.the (2008, main Editionresearch 4). methodologiesDiscovering Geometry: used An Investigativesomething Approach. that Emeryville: is only Key the Curriculum privilege Press.of some select by research mathematicians. In that sense, this mathematically endowed individuals. 7. Stieger, S. & Swami, V. (2015). Time to let go? No automatic aesthetic preference for the golden ratio in art pictures. Psychology of Aesthetics, Creativity, and the Arts, Vol 9(1), Feb, 91-100. http://dx.doi.org/10.1037/a0038506

8. Walser, H. (2001). The Golden Section. Washington, DC: The Mathematical Association of America. Reference: